Free space optical (FSO) communication systems are known for high-level capacity and information security. The overall system performances of FSO systems are however significantly affected by atmospheric turbulence induced fading. This paper, therefore, proposes a space-time code (STC) technique to mitigate this effect through the introduction of an additional degree of error correction capacity by exploiting the spectral dimension in the coding space. A space-time trellis coded orthogonal frequency division modulation (OFDM) scheme was developed, simulated and evaluated for free space optical communication through a gamma-gamma channel. The evaluation of the coding gain obtained from the simulation results, the mathematical analysis and the truncation error analysis shows that the proposed technique is a promising and viable technique for improving the error correction performance on optical communication links.

1. International Journal of Electrical and Computer Engineering (IJECE)
Vol. 13, No. 6, December 2023, pp. 6435~6442
ISSN: 2088-8708, DOI: 10.11591/ijece.v13i6.pp6435-6442  6435
Journal homepage: http://ijece.iaescore.com
Improving the performance of free space optical systems: a
space-time orthogonal frequency division modulation approach
Adedayo Olukayode Ojo, Isreal Esan Owolabi, Moses Oluwafemi Onibonoje
Department of Electrical/Electronics and Computer Engineering, College of Engineering, Afe Babalola University Ado Ekiti,
Ekiti State, Nigeria
Article Info ABSTRACT
Article history:
Received Oct 7, 2022
Revised Jun 6, 2023
Accepted Jun 26, 2023
Free space optical (FSO) communication systems are known for high-level
capacity and information security. The overall system performances of FSO
systems are however significantly affected by atmospheric turbulence induced
fading. This paper, therefore, proposes a space-time code (STC) technique to
mitigate this effect through the introduction of an additional degree of error
correction capacity by exploiting the spectral dimension in the coding space.
A space-time trellis coded orthogonal frequency division modulation
(OFDM) scheme was developed, simulated and evaluated for free space
optical communication through a gamma-gamma channel. The evaluation of
the coding gain obtained from the simulation results, the mathematical
analysis and the truncation error analysis shows that the proposed technique
is a promising and viable technique for improving the error correction
performance on optical communication links.
Keywords:
Coding gain
Coherent detection
Free space optical
communication
Gamma-gamma channel
Space-time trellis codes This is an open access article under the CC BY-SA license.
Corresponding Author:
Adedayo Olukayode Ojo
Department of Electrical/Electronic and Computer Engineering, College of Engineering, Afe Babalola
University Ado Ekiti
PMB 5454, Km 8.5, Afe Babalola Way, Ado Ekiti. Ekiti State, Nigeria
Email: ojoao@abuad.edu.ng
1. INTRODUCTION
Performance expectations for new generational communication systems are always increasing. This
is partly a consequence of users’ demand for high bandwidth, information security as well as high capacity.
These demands have thus far served as the impetus for the design of communication systems that more
efficiently meet users’ expectations [1]. In recent years, the radio frequency (RF) spectrum has come under
heavy bandwidth, data security, power and licensing constraints, so much that communication systems
designers are now looking to deploy optical communication solutions for high-speed communication links.
Optical fiber communication is one of the first solutions that became widely accepted for backbone
connectivity and other high-capacity point-to-point connections. However, its need for laborious cabling and
layout is shifting attention to free space optical (FSO) in an attempt to establish wireless transmission of
information in some applications. Free space optical communication entails the modulation of user information
onto optical carriers, and the onward transmission of same over unguided free space channel to photoreceptors.
FSO communication systems have proven to be up to this task by offering ultra-high data rates, enormous
bandwidth, high data security, high immunity to interference and jamming, and avoidance of regulatory issues
and spectrum licensing [2].
Broadly, FSO systems are often categorized based on their reception mechanisms. The most widely
reported type of FSO system is the intensity modulation/direct detection (IM/DD) [3]. In this type of FSO
communication system, the data to be transmitted is encoded by mapping the data signal to the intensity of the

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optical signal. At the receiving end, the variation in the irradiance intensity of the received optical signal is
simply used to represent the transmitted data [4] such that the converted AC photocurrent is proportional to the
optical signal power.
Even though FSO communication systems boast several advantages over their RF-based counterparts,
a significant drawback is that the performances of FSO-based communication systems are quite easily reduced
by turbulent conditions in the atmosphere [5]. The main culprit for this reduction in performance is the
inhomogeneity of the atmosphere as optical signals pass through, owing to varying thermal and pressure
gradients of the free space between optical lasers and the photoreceptors. Suspended particles, fog and droplets
are also factors that contribute to this reduction in the performance of FSO systems [6]. It has therefore become
necessary that solutions to this problem be devised. And one of such turbulence-mitigation solutions is the
adaptation of efficient coding schemes that have hitherto been quite successful in RF systems like transmit
antenna selection multiple-input multiple-output (MIMO) systems [7], turbo coded RF system [8], super-
orthogonal codes for Nakagami fading channels [9], binary phase-shift keying (BPSK) trellis coded orthogonal
frequency division modulation (OFDM) MIMO system [10], MIMO OFDM RF systems [11] and hybrid
concatenated codes with iterative decoding RF systems [12], for FSO systems. Other similar adaptable RF
error mitigating techniques include concatenated low-density parity-check (LDPC) space-time codes [13],
codeword diversity technique [14], and interleaving technique for high-speed OFDM systems [15], for FSO
systems as reported in space-time coded FSO system as reported in [16] and space-time coded OFDM FSO
system reported in [17]. Other turbulence mitigation schemes proposed so far include the exploitation of some
diversity properties to enhance the quality of communication by reducing the overall error on FSO links. Some
of these diversity schemes include temporal diversity, spatial diversity and spectral diversity [18]. By
implementing mechanisms to guarantee the delivery of multiple identical copies of a message to the receiver
through different paths and subchannels, these schemes have shown to significantly improve reliability of
communication over FSO links.
Several distributions have been used to model the free space optical communication channel with
different performances for different turbulence strengths. These include the log-normal distribution,
K-distribution, negative exponential distribution and gamma-gamma distribution. Of all these, the gamma-
gamma distribution possesses the advantage of being applicable for a wider range of turbulence, covering
weak turbulence, mild turbulence and strong turbulence [19]. In this paper, an approach for reducing the
unwanted results of atmospheric turbulence on the performance of FSO links is presented by exploiting the
spectral dimension of the coding space in addition to the coding gain and diversity offered by space-time
coding schemes.
2. RESEARCH METHOD
The approach adopted for this study include modelling of the free space optical channel under the
adopted distribution assumption and the development of the proposed space-time trellis coded OFDM coding
scheme. The conceptual framework for the proposed turbulence mitigation scheme was developed. The
developed scheme was thereafter deployed over the channel under the modelled conditions. The effectiveness
of the proposed approach at reducing errors was evaluated in relation to similar works. Finally, the performance
of the proposed scheme was validated with mathematical analysis. Modelling of communication channels using
statistical distributions is one of the approaches that have been deployed in depicting atmospheric turbulence
over the years. These distributions attempt to model the channel condition and behavior with the purpose of
understanding the impact of the channel on propagating optical signals.
Since interest in FSO systems has gathered pace over the last decade, several other variants of FSO
systems have been researched and reported. For example, coherent FSO systems are also gaining vast interest
[20] wherein the received signal is firstly coherently merged with a continuous wave local oscillator. Other
common forms of FSO systems include differential FSO systems [21], other variants of FSO systems
[22]–[25], and even recently, distributed FSO systems [19].
2.1. The system model
The Rayleigh fading model does not satisfactorily cater for the large-scale effect on propagating
optical signals, as such, proper modification is therefore required for application for FSO communication. Now,
assuming a unit energy signal, and considering that the overall atmospheric turbulence effect on the transmitted
signal is a combination of the small-scale effects, 𝐼𝑥, (also known as scattering) and the large-scale effect
(refraction), 𝐼𝑦. Such that the received normalized irradiance I is written as (1),
𝐼 = 𝐼𝑥 𝐼𝑦 (1)

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𝐼𝑥 and 𝐼𝑦 being gamma distribution functions themselves allow the gamma-gamma distribution to be modelled
as (2) [26]:
𝑝(𝐼𝑥) =
𝛼(𝛼𝐼𝑥)𝛼−1
Γ(𝛼)
𝑒(−𝛼𝐼𝑥 )
𝐼𝑥 > 0; 𝛼 > 0 (2)
and,
𝑝(𝐼𝑦) =
𝛽(𝛽𝐼𝑦)
𝛽−1
Γ(𝛽)
𝑒(−𝛽𝐼𝑦 )
𝐼𝑦 > 0; 𝛽 > 0. (3)
The conditional probability distribution function may consequently be written as seen in (4).
𝑝 (
𝐼
𝐼𝑥
) =
𝛽(
𝛽𝐼
𝐼𝑥
)
𝛽−1
𝐼𝑥Γ(𝛽)
𝑒
(−
𝛽𝐼
𝐼𝑥
)
𝐼 > 0 (4)
Averaging (4) over the range in order to obtain the unconditional distribution function, the gamma-gamma
distribution function for the channel irradiance is therefore obtained by (5):
𝑝(𝐼) = ∫ 𝑝 (
𝐼
𝐼𝑥
) 𝑝(𝐼𝑥)𝑑𝑥
∞
0
(5)
and expressed as (6) [27]:
𝑝(𝐼) =
2(𝛼𝛽)
(
𝛼+𝛽
2
)
Γ(𝛼)Γ(𝛽)
∙ 𝐼
(𝛼+𝛽)
2
−1
𝐾𝛼−𝛽(2√𝛼𝛽𝐼) 𝐼 > 0 (6)
where 𝐾𝛼−𝛽 is the modified Bessel function (also known as hyperbolic Bessel function) of the second kind of
order 𝛼 − 𝛽 , and 𝐼 is the irradiance. The variables 𝛼 and 𝛽 are the turbulence parameters expressed as functions
of the Roytov variance. The selection of the values of 𝛼 and 𝛽 is dependent on another entity called the Roytov
variance 𝜎𝑥
2
.
𝜎𝑥
2
= 1.33𝑘
7
6𝐶𝑛
2(𝐿)
11
6 (7)
However, the value of α is often greater than β and they both describe the physical effects of large-
scale and small-scale scintillations. Also, the gamma-gamma model was adopted in this study because of its
accuracy in modeling a wider range of turbulence conditions than other models such as log-normal distribution
or K-distribution. For the different turbulence types, the corresponding Roytov ranges are highlighted in
Table 1.
Table 1. Different turbulence types and their corresponding Roytov variance values
S/N Roytov range Turbulence type
1 𝜎 ≤ 0.3 Weak turbulence
2 0.3 < 𝜎 ≤ 5 Moderate turbulence
3 𝜎 > 5 Strong turbulence
Modifying the Bessel equations originally presented by [28], the modified Bessel function of the
second kind, 𝐾𝛼−𝛽, in the gamma-gamma relation in (6) is written in terms of the modified Bessel function of
the first kind 𝐼(𝛼−𝛽)(𝑥) as (8),
𝐾(𝛼−𝛽)(𝑥) =
𝜋
2
𝐼−(𝛼−𝛽)(𝑥)−𝐼(𝛼−𝛽)(𝑥)
sin(𝛼−𝛽)𝜋
(8)
where 𝐼(𝛼−𝛽)(𝑥) is expressed as (9).
𝐼(𝛼−𝛽)(𝑥) = ∑
1
𝑚!Γ(𝑚+(𝛼−𝛽)+1)
(
𝑥
2
)
2𝑚+(𝛼−𝛽)
∞
𝑚=0 (9)

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In order to simulate the effects of multipath fading in the channel, the simulation employs a tap model
approach with delay lines and different tap points. In the process, the total fading effect on the transmitted
signal was estimated. This estimation involves the computation of the total fading effect as a summation of the
total powers in the individual taps. Thereafter, space-time coded signals are transmitted into the channel and
the effect thereof is estimated upon arriving at the receiving end.
2.2. Space-time trellis code for free space optical communication link
Space-time trellis code (STTC) attempts to combine digital modulation schemes with trellis structures
in order to attain full rate, full diversity as well as coding gain. The possibility of adopting STTC for multiple
antennas is of particular interest to this work. This enables the transmission of information over multiple laser
diodes through free space. In other words, space-time trellis codes are STCs adapted for multiple transmit laser
diodes.
The fundamental purpose of a space-time trellis encoder is to utilize mapping functions that serve as
representations of their corresponding trellis diagrams for mapping binary data to modulation symbols. In this
study, we employ the specific space-time trellis code depicted in Efendi [29]. To ensure simplicity while
preserving the desired MIMO configuration, two laser diodes were utilized. Consequently, the input bit stream
c for the space-time trellis code, taking into account the presence of the two laser diodes, is expressed as (10).
𝑐 = (𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑡, … ) (10)
The variable 𝑐𝑡 represents a pair of information bits at a specific time instant, as defined in (11). By
utilizing feedforward shift registers, the encoder converts the input bit array into a sequence of QPSK
modulated signals, denoted as 𝑥0, 𝑥1, 𝑥2, …, and so on. Each element in this array corresponds to the space-
time symbol at a particular time t.
𝑐1 = (𝑐𝑡
1
, 𝑐𝑡
2) (11)
The output 𝑥𝑡
𝑖
of the encoder for the 𝑖-th transmitter at time t is expressed, as introduced in [30], in
(12). Thus, the STTC encoder translates the input bit stream into quadrature phase shift keying (QPSK)
modulated symbols.
𝑥𝑡
𝑖
= ∑ ∑ 𝑔𝑗,𝑖
𝑘
𝑐𝑡−𝑗
𝑘
Mod 2
𝑣𝑘
𝑗=0
𝑚
𝑘=1 , 𝑖 = 1, 2 (12)
The encoding structure and the decoding structure are illustrated in Figure 1. At the transmit end, the
input bits are encoded according to the specified trellis structure and the generator matrix, the codewords are
then transmitted into free space through the transmit lasers onto the receivers. At the other end of the system,
the optical signal being received is combined and detected, and is forwarded for onward decoding using Viterbi
algorithm. The original transmitted bit is thereby recovered and the error therein is evaluated.
Figure 1. The encoding and decoding structure of the STTC FSO communication system
The parameters involved in the simulation of the space-time trellis coded free space optical
communication system were as specified. The effect of different trellis structures of different states was
investigated for this proposed turbulence mitigating scheme. These include the 16-state trellis structure and the
32-state trellis structure. The QPSK modulation scheme was integrated with the trellis structure. The
parameters involved in the simulation of the space-time trellis coded free space optical communication system
include a frame size of 10,000, a modulation type of QPSK at full rate, two laser diodes and one photoreceptor.

5. Int J Elec & Comp Eng ISSN: 2088-8708 
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2.3. Truncation error analysis
Owing to the use of different mathematical series (binomial expansion and Taylors series) in the
mathematical error rate analysis, a certain approximation error is introduced as a result of the truncations taken
at various terms in other to make the solutions finite. This section attempts to estimate this error. If we represent
the truncation error resulting from discarding all terms after the first J+1 terms as (13).
𝜀𝐽,𝑀 = ∑ (
2𝐿
𝑖
) ∑ 𝑢𝑝(𝛼, 𝛽, 𝑖, 𝑀) (
1
𝜇𝑀𝛾
̅
)
𝑝
∞
𝑝=𝐽+1
2𝐿
𝑖=0 (13)
The function 𝑢𝑝(. , . , . , . ) is expounded into (14). The Taylor series expansion was combined with the
identity expressed in (20). The limit analysis on the truncations error, the upper bound for the truncation error
can therefore be expressed using (14) to (17).
𝑢𝑝(𝑥, 𝑦, 𝑖, 𝑀) =
𝜑(𝑝+2𝑦+𝑖(𝑥−𝑦),
𝑀−1
𝑀
)
𝜋
× 𝑏𝑝(2𝐿 − 𝑖, 𝑖, 𝑥, 𝑦) (
1
𝜇𝑀𝛾
̅
)
2𝑦+𝑖(𝑥−𝑦)
(14)
𝑥𝐽+1
(1−𝑥)
= 𝑥𝐽+1
+ 𝑥𝐽+2
+ 𝑥𝐽+3
+ ⋯ + 𝑥𝐽+𝑛+1
+ ⋯, (15)
(
2𝐿
𝑖
) =
2𝐿
𝑖!(2𝐿−1)!
(16)
𝜀𝐽,𝑀 ≤
2
𝜋(𝜇𝑀𝛾
̅−1)
(
1
𝜇𝑀𝛾
̅
)
𝐽
∑
𝑚𝑎𝑥
𝑝>𝐽 {𝑢𝑝(𝛼,𝛽,𝑖,𝑀)}
𝑖!(2𝐿−1)!
2𝐿
𝑖=0 1 (17)
3. RESULTS AND ANALYSIS
The effect of space-time trellis code (STTC) in mitigating turbulence induced fading on free space
optical communication links was investigated. This was done for weak turbulence, moderate turbulence and
for strong turbulence. The performances of the space-time trellis coded FSO system under the above turbulence
strength were evaluated against the uncoded FSO system and the results thereof are plotted in Figures 2(a)
and 2(b).
(a) (b)
Figure 2. The performance of 16-State STTC for FSO link (a) under different turbulence strengths and (b) in
comparison with performance from literature
The 16-state space-time trellis code performed well when compared with the uncoded system
throughout the range of SNR. This performance of the space-time trellis coded FSO system becomes evident
at higher SNR values where, at bit error rate of 10-8
for example, a coding gain of 5 dB was attained over the
uncoded system. To further evaluate the performance of the STTC FSO system reported in Figure 2, the

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performances of the space-time trellis coded FSO system in this study were compared against a similar
space-time trellis coded system as reported in [31]. To facilitate logical cross-comparison, this performance
comparison was carried out over strong atmospheric turbulence. The STTC FSO code in the study slightly
outperforms their reported system. At a bit error rate of 10-7
for instance, the STTC FSO in this study attains a
coding gain of about 2 dB over the performance reported in [31]. This further underscores the importance of
coding schemes such as the one proposed in this work and [32]–[35].
4. CONCLUSION
In this study, a space-time coded OFDM scheme has been developed for free space optical
communication under gamma-gamma assumptions. Achieved by exploiting the spectral dimension through the
integration of OFDM, the technique was able to mitigate the effect of turbulence-induced fading on transmitted
optical signals as they propagate through free space. In addition, the proposed technique represents a way to
enable us rapidly evaluate the channel behavior of gamma-gamma FSO links for space-time coded data
transmission. The results obtained show easily and rapidly, the behavior of MIMO FSO links under given
average SNR and turbulence conditions. This in-depth understanding of the channel response to different
parameters affords designers the opportunity to design improved and efficient coding schemes to facilitate low
error rates high speed wireless communication with enhanced reliability. All these are tailored towards
realizing the full potential of FSO systems as an attractive solution to last-mile problems. In conclusion, free
space optical communication offers numerous advantages as a promising technology. However, the presence
of atmospheric turbulence poses a significant obstacle that must be overcome to fully harness the potential of
this communication method. This article has presented a solution to tackle this challenge. By addressing the
issue of atmospheric turbulence, we can pave the way for improved performance and reliability in free space
optical communication systems. Through ongoing research and development, we can further refine and
enhance this proposed approach, ultimately unlocking the full benefits of free space optical communication in
various applications.
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BIOGRAPHIES OF AUTHORS
Adedayo Olukayode Ojo graduated from the Department of Electronic and
Electrical Engineering in Ladoke Akintola University of Technology (LAUTECH) Nigeria
where he received his B. Tech degree with honours. He obtained his M.Sc. degree in
Microelectronics from Universiti Putra Malaysia (UPM). He is currently a lecturer at the
Department of Electronic Electrical and Computer Engineering, Afe Babalola University Ado
Ekiti, Ekiti State, Nigeria. He is currently pursuing a Ph.D. degree at the Department of
Electrical/Electronics and Computer Engineering, Afe Babalola University Ado-Ekiti, Nigeria.
His research interests include communication systems, microelectronics, soft computing and
intelligent systems. He can be contacted at email: ojoao@abuad.edu.ng.

8.  ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 13, No. 6, December 2023: 6435-6442
6442
Isreal Esan Owolabi received his bachelor of science degree from the University
of Ibadan, Nigeria in 1961 and his Ph.D. from University of Durham, England in 1964. A
professor of Electrical Electronic Engineering at Afe Babalola University Ado-Ekiti Nigeria and
a fellow of the Institute of Electrical Electronic Engineers, he is the patron of the Center for
Research in Electrical Communication (CRECO), Ekiti State University, Ado-Ekiti, Nigeria.
His research interests are in the areas of communication systems, signal propagation,
antenna and radar systems, and electromagnetics. He can be contacted at email:
eowolabi2002@yahoo.com.
Moses Oluwafemi Onibonoje is an Associate Professor in the Department of
Electrical/Electronics and Computer Engineering at Afe Babalola University Ado-Ekiti
(ABUAD), Nigeria. He specializes in control and instrumentation engineering. His research
areas include wireless sensor networks, distributed systems, wireless instrumentations,
optimization methods, and agro-electro instrumentations. Dr. Onibonoje holds a Ph.D. in
Electronic and Electrical Engineering from Obafemi Awolowo University, Ile-Ife, Nigeria. He
is COREN registered and a corporate member of the Nigerian Society of Engineers. He can be
contacted at email: onibonojemo@abuad.edu.ng.